Transcript Shell model

16.451: Yesterday’s news!
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16.451 Lecture 22: Beyond the mass formula...
Solid line: fit to the
semi-empirical formula
some large
oscillations
at small mass
25/11/2003
almost flat, apart from Coulomb effects
Most stable: 56Fe,
8.8 MeV/ nucleon
very sharp rise
at small A
gradual decrease
at large A due to
Coulomb repulsion
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Semi-Empirical Mass (binding energy) Formula (SEMF) implications:
B(Z , A)
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 aV A  aS A2 / 3 aC Z (Z 1) A1/ 3  a A ( A  2Z ) 2 A1  
Stable nuclei have the maximum B for a given A; for constant mass number,
B is quadratic in Z  “mass parabolas”, e.g.:
Even A: offset
is the pairing term!
Beyond the SEMF: “Magic Numbers” and the Shell Model
We already noted that there
were some marked deviations
from the SEMF curve at small
mass number, e.g. A = 4.
On an enlarged scale, a systematic
pattern of deviations occurs, with
maxima in B occurring for certain
“magic” values of N and Z, given by:
N/Z = 2, 8, 20, 28, 50, 82, 126
These values of neutron and proton
number are anomalously stable
with respect to the average – the
pattern must therefore reflect
something important about the
average nuclear potential V(r) that
the neutrons and protons are
bound in....
(NB, the most stable nucleus of all is
56Fe, which has Z = 28, N = 28,
“double magic” ...)
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Other evidence for “magic numbers” 2-n and 2-p separation energies:
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• Energy required to remove a pair of
neutrons or protons from a given nucleus is
referred to as S2n or S2p
• like the ionization energy for atoms, but
the pairing force is so strong in nuclei that
systematics are more easily seen comparing
nuclei that differ by 2 nucleons
• the same pattern of “magic numbers”
appears – large separation energies
correspond to particularly stable nuclei:
N/Z = 2, 8, 20, 28, 50, 82, 126 ....
A periodic table of nuclei?
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Systematics are reminiscent of the periodic structure of atoms, which results from
filling independent single-particle electron states with electrons in the most efficient
way consistent with the Pauli principle, but the magic numbers are different:
“Magic numbers” for atoms:
Z = 2, 10, 18, 30, 36, 48, 54, 70, 80, 86 ....
Single Particle Shell Model for Nuclei:
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Self-consistent approximation: assume the quantum state of the ith nucleon can be found
by solving a Schrödinger equation for its interaction via an average nuclear potential VN(r)
due to the other (A-1) nucleons:
  2 2
  VN (r )

 2




(
r
)

E

(
r
 nlm i
nl nlm i )

Q
Assume a spherically symmetric potential VN(r);
then the eigenstates have definite orbital angular
momentum, and the standard radial and angular
momentum quantum numbers (n,l,m) as indicated.
(Justification: measured quadrupole moments of
nuclei are relatively small, at least near the
“magic numbers” that we are interested in explaining;
midway between the last two magic numbers, ie
around Z or N = 70, 100, the picture changes, and
we will have to use a different approach, but at
least for the lighter nuclei this assumption should
be reasonable.)
A
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Shell Model continued...
If we choose the right potential function VN(r), then the wave function for the whole
nucleus can be written as a product of the single particle wave functions for all A
nucleons, or at least schematically:


Nucleus (r ) 
A
i 1

 nlm (ri )
oversimplification here... actually, it has to be written as an
antisymmetrized product wavefunction since the nucleons are
identical Fermions – the procedure is well-documented in
advanced textbooks in any case!
With total angular momentum given by:

J 
A


ji ,



ji  i  si ,
i 1
And parity:
 

A
i 1
(1)
i
(s 
1
2
)
Always + for an even
number of nucleons...
What to use for VN(r)? – three candidate potential functions:
Advantage: easy to write down
Disadvantages:
numerical solutions only
edges unrealistically sharp
Advantage: easy to write down and
can be solved analytically
Disadvantage: potential should
not go to infinity, have to cut off
the function at some finite r and
adjust parameters to fit data.
Advantage: same shape as measured
charge density distributions of
nuclei. smooth edge makes sense
Disadvantage:
numerical solution needed
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Comparison: Harmonic Oscillator versus Woods-Saxon solutions:
• since both potentials are spherically
symmetric, the only difference is in
the radial dependence of the wave
functions
• amazingly, when parameters are
adjusted to make the average
potential the same, as shown in the
top panel, there is remarkably
little difference in the radial
probability densities for these
two potential energy functions!
• this being the case, the simplicity
of the harmonic oscillator potential
means that it is strongly preferred
as a model for nuclei
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Evidence that this works: (Krane, Fig. 5.13)
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electric charge density, measured via electron scattering:
charge density difference
between 205Tl and 206 Pb is
proportional to the square
of the wave function for
the extra proton in 206 Pb,
i.e. we can actually measure
the square of the wave
function for a single proton
in a complex nucleus this
way!

 (r )  e
Z


|  i (r ) | 2
i 1
Theory:
square of the harmonic
oscillator wave function
for the last proton in 206Pb,
quantum numbers: n = 3, l = 0
--- it works !!!
Various potential shapes lead to similar patterns of energy gaps, e.g.:
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But the magic numbers are wrong  !
N/Z = 2, 8, 20, 28, 50, 82, 126
Something else is needed to explain
the observed behaviour...
Solution: the “spin-orbit” potential
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• Meyer and Jensen, 1949: enormous breakthrough at the time because it was the
only explanation for the observed pattern of “magic numbers” and paved the way
for a “periodic table” of nuclei ... and the Nobel prize in physics, 1963!
• simple idea:
(http://www.nobel.se/physics/laureates/1963/index.html
– see Maria Goeppert-Meyer’s Nobel Lecture link on this page)
 
VN (r )  VN (r )  Vso (r )   s
as in the calculation of magnetic moments, lecture 19, we can write:
 
s 
1
2
j 2  2  s2 
1
2
 j ( j  1)  (  1)  s(s  1)
• but there are only two ways the orbital and spin angular momentum can add for a
single particle nucleon state:
a) “stretched state”
j=l+½:
b) “jack-knife state” j = l – ½:
 

  s  s  2
 
(  1)
  s   s (  1) 
2
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Spin-orbit force, continued:
• The energy shift due to the spin-orbit interaction is between states of the same
l but different j;
n, j    1 / 2
n, l
E
n, j    1 / 2
• the splitting is proportional to l and so it increases as the energy increases for the
single particle solutions to V(r)
• each state can accommodate (2j+1) neutrons or protons, each with different mj
• empirically, the sign of the spin-orbit term for nuclei is opposite to that for atoms
and the effect is much stronger in nuclei – the phenomenon has nothing to do with
magnetism, which is the origin of this effect in atoms, but rather it reflects a basic
feature of the strong nuclear force.
• with these features, the spin-orbit potential is the “missing link” required to
correctly predict the observed sequence of magic numbers in nuclear physics 
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something to read.... http://www.nobel.se/physics/laureates/1963/
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see also!
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